Theorem tpid3g 3722

Description: Closed theorem form of tpid3 3723. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
tpid3g	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2766	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
2		3mix3 1170	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
3	2	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)))
4		abid 2177	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
5	3, 4	imbitrrdi 162	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}))
6		dftp2 3656	. . . . . 6 ⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}
7	6	eleq2i 2256	. . . . 5 ⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)})
8	5, 7	imbitrrdi 162	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9		eleq1 2252	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
10	8, 9	mpbidi 151	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
11	10	exlimdv 1830	. 2 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
12	1, 11	mpd 13	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 979   = wceq 1364  ∃wex 1503   ∈ wcel 2160  {cab 2175  {ctp 3609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615

This theorem is referenced by:  rngmulrg  12652  srngmulrd  12663  lmodscad  12681  ipsmulrd  12693  ipsipd  12696  topgrptsetd  12713